BGE: How do I make a sensor of one object trigger an actuator of another object?
Add a collision sensor in the button. There are 2 options there:


collision with the material of the player and
collision with a property that you can add to a player.


Then add an action actuator to the door. Remember that you should add the animation actuator to the object that is gonna get animated.

At last, connect them by dragging the "noodle" from the end of the sensor of object 1 to the start of the actuator of object 2. Remember to have them both selected so you can see both their SCAs.

In case those 2 can't be connected add manually a controller "And" and drag the noodle from the sensor to the start of the controller, and then drag the noodle from the end of the "And: controller to the start of the actuator.